Algorithmic Motion Planning and Related Geometric Problems on Parallel Machines (Dissertation Proposal)

The problem of algorithmic motion planning is one that has received considerable attention in recent years. The automatic planning of motion for a mobile object moving amongst obstacles is a fundamentally important problem with numerous applications in computer graphics and robotics. Numerous approximate techniques (AI-based, heuristics-based, potential field methods, for example) for motion planning have long been in existence, and have resulted in the design of experimental systems that work reasonably well under various special conditions [7, 29, 30]. Our interest in this problem, however, is in the use of algorithmic techniques for motion planning, with provable worst case performance guarantees. The study of algorithmic motion planning has been spurred by recent research that has established the mathematical depth of motion planning. Classical geometry, algebra, algebraic geometry and combinatorics are some of the fields of mathematics that have been used to prove various results that have provided better insight into the issues involved in motion planning [49]. In particular, the design and analysis of geometric algorithms has proved to be very useful for numerous important special cases. In the remainder of this proposal we will substitute the more precise term of algorithmic motion planning by just motion planning

Convex Hulls: Complexity and Applications (a Survey)

Computational geometry is, in brief, the study of algorithms for geometric problems. Classical study of geometry and geometric objects, however, is not well-suited to efficient algorithms techniques. Thus, for the given geometric problems, it becomes necessary to identify properties and concepts that lend themselves to efficient computation. The primary focus of this paper will be on one such geometric problems, the Convex Hull problem

Optimal Parallel Randomized Algorithms for the Voronoi Diagram of Line Segments in the Plane and Related Problems

In this paper, we present an optimal parallel randomized algorithm for the Voronoi diagram of a set of n non-intersecting (except possibly at endpoints) line segments in the plane. Our algorithm runs in O(log n) time with very high probability and uses O(n) processors on a CRCW PRAM. This algorithm is optimal in terms of P.T bounds since the sequential time bound for this problem is Ω(n log n). Our algorithm improves by an O(log n) factor the previously best known deterministic parallel algorithm which runs in O(log2 n) time using O(n) processors [13]. We obtain this result by using random sampling at two stages of our algorithm and using efficient randomized search techniques. This technique gives a direct optimal algorithm for the Voronoi diagram of points as well (all other optimal parallel algorithms for this problem use reduction from the 3-d convex hull construction)

Optimal Mesh Algorithms for the Voronoi Diagram of Line Segments, Visibility Graphs and Motion Planning in the Plane

The motion planning problem for an object with two degrees of freedom moving in the plane can be stated as follows: Given a set of polygonal obstacles in the plane, and a two-dimensional mobile object B with two degrees of freedom, determine if it is possible to move B from a start position to a final position while avoiding the obstacles. If so, plan a path for such a motion. Techniques from computational geometry have been used to develop exact algorithms for this fundamental case of motion planning. In this paper we obtain optimal mesh implementations of two different methods for planning motion in the plane. We do this by first presenting optimal mesh algorithms for some geometric problems that, in addition to being important substeps in motion planning, have numerous independent applications in computational geometry. In particular, we first show that the Voronoi diagram of a set of n nonintersecting (except possibly at endpoints) line segments in the plane can be constructed in O(√n) time on a √n x √n mesh, which is optimal for the mesh. Consequently, we obtain an optimal mesh implementation of the sequential motion planning algorithm described in [14]; in other words, given a disc B and a polygonal obstacle set of size n, we can plan a path (if it exists) for the motion of B from a start position to a final position in O (√n) time on a mesh of size n. Next we show that given a set of n line segments and a point p, the set of segment endpoints that are visible from p can be computed in O (√n) mesh-optimal time on a √n x √n mesh. As a result, the visibility graph of a set of n line segments can be computed in O(n) time on an n x n mesh. This result leads to an O(n) algorithm on an n x n mesh for planning the shortest path motion between a start position and a final position for a convex object B (of constant size) moving among convex polygonal obstacles of total size n

Quadrilateral Meshes with Bounded Minimum Angle

This paper presents an algorithm that utilizes a quadtree to construct a strictly convex quadrilateral mesh for a simple polygonal region in which no newly created angle is smaller than . This is the first known result, to the best of our knowledge, on quadrilateral mesh generation with a provable guarantee on the minimum angle

Quadrilateral Meshes with Bounded Minimum Angle

This paper presents an algorithm that utilizes a quadtree to construct a strictly convex quadrilateral mesh for a simple polygonal region in which no newly created angle is smaller than . This is the first known result, to the best of our knowledge, on quadrilateral mesh generation with a provable guarantee on the minimum angle

An O(n log n)-Time Algorithm for the Restricted Scaffold Assignment

The assignment problem takes as input two finite point sets S and T and establishes a correspondence between points in S and points in T, such that each point in S maps to exactly one point in T, and each point in T maps to at least one point in S. In this paper we show that this problem has an O(n log n)-time solution, provided that the points in S and T are restricted to lie on a line (linear time, if S and T are presorted).Comment: 13 pages, 8 figure